Rework of GenericMinMax: new axiomatic, split logical/decidable parts, Leibniz part

 Moreover, instantiation like MinMax are now made without redefining
 generic properties (easier maintenance). We start using inner modules
 for qualifying (e.g. Z.max_comm).

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12638 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 20 insertions, 118 deletions

diff --git a/theories/Arith/MinMax.v b/theories/Arith/MinMax.v
index caf860af8..32afba489 100644
--- a/theories/Arith/MinMax.v
+++ b/theories/Arith/MinMax.v
@@ -26,138 +26,40 @@ Fixpoint min n m : nat :=
 
 (** These functions implement indeed a maximum and a minimum *)
 
-Lemma max_spec : forall x y,
-  (x<y /\ max x y = y)  \/  (y<=x /\ max x y = x).
+Lemma max_l : forall x y, y<=x -> max x y = x.
 Proof.
  induction x; destruct y; simpl; auto with arith.
- destruct (IHx y); intuition.
 Qed.
 
-Lemma min_spec : forall x y,
-  (x<y /\ min x y = x)  \/  (y<=x /\ min x y = y).
+Lemma max_r : forall x y, x<=y -> max x y = y.
 Proof.
  induction x; destruct y; simpl; auto with arith.
- destruct (IHx y); intuition.
 Qed.
 
-Module NatHasMinMax <: HasMinMax Nat_as_OT.
- Definition max := max.
- Definition min := min.
- Definition max_spec := max_spec.
- Definition min_spec := min_spec.
-End NatHasMinMax.
-
-(** We obtain hence all the generic properties of max and min. *)
-
-Module Import NatMinMaxProps := MinMaxProperties Nat_as_OT NatHasMinMax.
-
-(** For some generic properties, we can have nicer statements here,
-    since underlying equality is Leibniz. *)
-
-Lemma max_case_strong : forall n m (P:nat -> Type),
-  (m<=n -> P n) -> (n<=m -> P m) -> P (max n m).
-Proof. intros; apply max_case_strong; auto. congruence. Defined.
-
-Lemma max_case : forall n m (P:nat -> Type),
-  P n -> P m -> P (max n m).
-Proof. intros. apply max_case_strong; auto. Defined.
-
-Lemma max_monotone: forall f,
- (Proper (le ==> le) f) ->
- forall x y, max (f x) (f y) = f (max x y).
-Proof. intros; apply max_monotone; auto. congruence. Qed.
-
-Lemma min_case_strong : forall n m (P:nat -> Type),
-  (n<=m -> P n) -> (m<=n -> P m) -> P (min n m).
-Proof. intros; apply min_case_strong; auto. congruence. Defined.
-
-Lemma min_case : forall n m (P:nat -> Type),
-  P n -> P m -> P (min n m).
-Proof. intros. apply min_case_strong; auto. Defined.
-
-Lemma min_monotone: forall f,
- (Proper (le ==> le) f) ->
- forall x y, min (f x) (f y) = f (min x y).
-Proof. intros; apply min_monotone; auto. congruence. Qed.
-
-Lemma max_min_antimonotone : forall f,
- Proper (le==>ge) f ->
- forall x y, max (f x) (f y) == f (min x y).
+Lemma min_l : forall x y, x<=y -> min x y = x.
 Proof.
- intros. apply max_min_antimonotone. congruence.
- intros z z' Hz; red; unfold ge in *; auto.
+ induction x; destruct y; simpl; auto with arith.
 Qed.
 
-Lemma min_max_antimonotone : forall f,
- Proper (le==>ge) f ->
- forall x y, min (f x) (f y) == f (max x y).
+Lemma min_r : forall x y, y<=x -> min x y = y.
 Proof.
- intros. apply min_max_antimonotone. congruence.
- intros z z' Hz; red; unfold ge in *; auto.
+ induction x; destruct y; simpl; auto with arith.
 Qed.
 
-(** For the other generic properties, we make aliases,
-   since otherwise SearchAbout misses some of them
-   (bad interaction with an Include).
-   See GenericMinMax (or SearchAbout) for the statements. *)
-
-Definition max_spec_le := max_spec_le.
-Definition max_dec := max_dec.
-Definition max_unicity := max_unicity.
-Definition max_unicity_ext := max_unicity_ext.
-Definition max_id := max_id.
-Notation max_idempotent := max_id (only parsing).
-Definition max_assoc := max_assoc.
-Definition max_comm := max_comm.
-Definition max_l := max_l.
-Definition max_r := max_r.
-Definition le_max_l := le_max_l.
-Definition le_max_r := le_max_r.
-Definition max_le := max_le.
-Definition max_le_iff := max_le_iff.
-Definition max_lt_iff := max_lt_iff.
-Definition max_lub_l := max_lub_l.
-Definition max_lub_r := max_lub_r.
-Definition max_lub := max_lub.
-Definition max_lub_iff := max_lub_iff.
-Definition max_lub_lt := max_lub_lt.
-Definition max_lub_lt_iff := max_lub_lt_iff.
-Definition max_le_compat_l := max_le_compat_l.
-Definition max_le_compat_r := max_le_compat_r.
-Definition max_le_compat := max_le_compat.
-
-Definition min_spec_le := min_spec_le.
-Definition min_dec := min_dec.
-Definition min_unicity := min_unicity.
-Definition min_unicity_ext := min_unicity_ext.
-Definition min_id := min_id.
-Notation min_idempotent := min_id (only parsing).
-Definition min_assoc := min_assoc.
-Definition min_comm := min_comm.
-Definition min_l := min_l.
-Definition min_r := min_r.
-Definition le_min_l := le_min_l.
-Definition le_min_r := le_min_r.
-Definition min_le := min_le.
-Definition min_le_iff := min_le_iff.
-Definition min_lt_iff := min_lt_iff.
-Definition min_glb_l := min_glb_l.
-Definition min_glb_r := min_glb_r.
-Definition min_glb := min_glb.
-Definition min_glb_iff := min_glb_iff.
-Definition min_glb_lt := min_glb_lt.
-Definition min_glb_lt_iff := min_glb_lt_iff.
-Definition min_le_compat_l := min_le_compat_l.
-Definition min_le_compat_r := min_le_compat_r.
-Definition min_le_compat := min_le_compat.
-
-Definition min_max_absorption := min_max_absorption.
-Definition max_min_absorption := max_min_absorption.
-Definition max_min_distr := max_min_distr.
-Definition min_max_distr := min_max_distr.
-Definition max_min_modular := max_min_modular.
-Definition min_max_modular := min_max_modular.
-Definition max_min_disassoc := max_min_disassoc.
+
+Module NatHasMinMax <: HasMinMax Nat_as_OT.
+ Definition max := max.
+ Definition min := min.
+ Definition max_l := max_l.
+ Definition max_r := max_r.
+ Definition min_l := min_l.
+ Definition min_r := min_r.
+End NatHasMinMax.
+
+(** We obtain hence all the generic properties of [max] and [min],
+    see file [GenericMinMax] or use SearchAbout. *)
+
+Module Export MMP := UsualMinMaxProperties Nat_as_OT NatHasMinMax.
 
 
 (** * Properties specific to the [nat] domain *)